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Source file src/math/atan.go

     1	// Copyright 2009 The Go Authors. All rights reserved.
     2	// Use of this source code is governed by a BSD-style
     3	// license that can be found in the LICENSE file.
     4	
     5	package math
     6	
     7	/*
     8		Floating-point arctangent.
     9	*/
    10	
    11	// The original C code, the long comment, and the constants below were
    12	// from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
    13	// http://www.netlib.org/cephes/cmath.tgz.
    14	// The go code is a version of the original C.
    15	//
    16	// atan.c
    17	// Inverse circular tangent (arctangent)
    18	//
    19	// SYNOPSIS:
    20	// double x, y, atan();
    21	// y = atan( x );
    22	//
    23	// DESCRIPTION:
    24	// Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
    25	//
    26	// Range reduction is from three intervals into the interval from zero to 0.66.
    27	// The approximant uses a rational function of degree 4/5 of the form
    28	// x + x**3 P(x)/Q(x).
    29	//
    30	// ACCURACY:
    31	//                      Relative error:
    32	// arithmetic   domain    # trials  peak     rms
    33	//    DEC       -10, 10   50000     2.4e-17  8.3e-18
    34	//    IEEE      -10, 10   10^6      1.8e-16  5.0e-17
    35	//
    36	// Cephes Math Library Release 2.8:  June, 2000
    37	// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
    38	//
    39	// The readme file at http://netlib.sandia.gov/cephes/ says:
    40	//    Some software in this archive may be from the book _Methods and
    41	// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
    42	// International, 1989) or from the Cephes Mathematical Library, a
    43	// commercial product. In either event, it is copyrighted by the author.
    44	// What you see here may be used freely but it comes with no support or
    45	// guarantee.
    46	//
    47	//   The two known misprints in the book are repaired here in the
    48	// source listings for the gamma function and the incomplete beta
    49	// integral.
    50	//
    51	//   Stephen L. Moshier
    52	//   moshier@na-net.ornl.gov
    53	
    54	// xatan evaluates a series valid in the range [0, 0.66].
    55	func xatan(x float64) float64 {
    56		const (
    57			P0 = -8.750608600031904122785e-01
    58			P1 = -1.615753718733365076637e+01
    59			P2 = -7.500855792314704667340e+01
    60			P3 = -1.228866684490136173410e+02
    61			P4 = -6.485021904942025371773e+01
    62			Q0 = +2.485846490142306297962e+01
    63			Q1 = +1.650270098316988542046e+02
    64			Q2 = +4.328810604912902668951e+02
    65			Q3 = +4.853903996359136964868e+02
    66			Q4 = +1.945506571482613964425e+02
    67		)
    68		z := x * x
    69		z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
    70		z = x*z + x
    71		return z
    72	}
    73	
    74	// satan reduces its argument (known to be positive)
    75	// to the range [0, 0.66] and calls xatan.
    76	func satan(x float64) float64 {
    77		const (
    78			Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
    79			Tan3pio8 = 2.41421356237309504880      // tan(3*pi/8)
    80		)
    81		if x <= 0.66 {
    82			return xatan(x)
    83		}
    84		if x > Tan3pio8 {
    85			return Pi/2 - xatan(1/x) + Morebits
    86		}
    87		return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
    88	}
    89	
    90	// Atan returns the arctangent, in radians, of x.
    91	//
    92	// Special cases are:
    93	//      Atan(±0) = ±0
    94	//      Atan(±Inf) = ±Pi/2
    95	func Atan(x float64) float64
    96	
    97	func atan(x float64) float64 {
    98		if x == 0 {
    99			return x
   100		}
   101		if x > 0 {
   102			return satan(x)
   103		}
   104		return -satan(-x)
   105	}
   106	

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